My question:
If a simulation produces 200 random samples of size 37 from the same population, then
the mean of these 200 samples will be equal to the population mean. the sample means will all be the same. the mean of these 200 samples will likely be close to the population mean. the sample variances will be the same.
I'm guessing that it is sample variances will be the same. But i'm not too sure how to work out if it is correct.
Thank you for your time! :)
The sampling distributions of sample means has the same mean as the parent distribution, but the variance is $\frac{\sigma ^2}{n}$, where $n$ is the sample size.
This is because $$Var(\bar X)=Var(\frac{X_1+X_2+...+X_n}{n})=\frac{1}{n^2}Var(X_1+X_2+...+X_n)$$
$$=\frac{1}{n^2}n\sigma^2=\frac{\sigma^2}{n}$$
You can prove that the mean is same as the parent distribution's mean with the same technique and using expectation algebra to find $E(\bar X)$